Inverse method to estimate the properties of a flexural beam and the corresponding boundary parameters

ABSTRACT

A system and method is used for estimating the properties of a flexural beam. The beam is shaken transverse to its longitudinal axis. Seven frequency domain transfer functions of displacement are measured at spaced apart locations along the beam. The seven transfer functions are combined to yield closed form values of the flexural wavenumber in propagation coefficients at any test frequency.

STATEMENT OF GOVERNMENT INTEREST

[0001] The invention described herein may be manufactured and used by orfor the Government of the United States of America for governmentalpurposes without the payment of any royalties thereon or therefore.

BACKGROUND OF THE INVENTION

[0002] (1) Field of the Invention

[0003] This invention relates to the field of structural properties, andin particular to the determination of the complex flexural wavenumber,corresponding wave propagation coefficients, and boundary conditionparameters of a beam subjected to transverse motion.

[0004] (2) Description of the Prior Art

[0005] By way of example of the state of the art, reference is made tothe following papers, which are incorporated herein by reference. Notall of these references may be deemed to be relevant prior art.

[0006] D. M. Norris, Jr., and W. C. Young, “Complex Modulus Measurementsby Longitudinal Vibration Testing,” Experimental Mechanics, Volume 10,1970, pp. 93-96.

[0007] W. M. Madigosky and G. F. Lee, “Improved Resonance Technique forMaterials Characterization,” Journal of the Acoustical Society ofAmerica, Volume 73, Number 4, 1983, pp. 1374-1377.

[0008] S. L. Garrett, “Resonant Acoustic Determination of ElasticModuli,” Journal of the Acoustical Society of America, Volume 88, Number1, 1990, pp. 210-220.

[0009] I. Jimeno-Fernandez, H. Uberall, W. M. Madigosky, and R. B.Fiorito, “Resonance Decomposition for the Vibratory Response of aViscoelastic Rod,” Journal of the Acoustical Society of America, Volume91, Number 4, Part 1, April 1992, pp. 2030-2033.

[0010] G. F. Lee and B. Hartmann, “Material Characterizing System,” U.S.Pat. No. 5,363,701, Nov. 15, 1994.

[0011] G. W. Rhodes, A. Migliori, and R. D. Dixon, “Method for ResonantMeasurement,” U.S. Pat. No. 5,495,763, Mar. 5, 1996.

[0012] R. F. Gibson and E. O. Ayorinde, “Method and Apparatus forNon-Destructive Measurement of Elastic Properties of StructuralMaterials,” U.S. Pat. No. 5,533,399, Jul. 9, 1996.

[0013] B. J. Dobson, “A Straight-Line Technique for Extracting ModalProperties From Frequency Response Data,” Mechanical Systems and SignalProcessing, Volume 1, 1987, pp. 29-40.

[0014] C. Minas and D. J. Inman, “Matching Finite Element Models toModal Data,” Journal of Vibration and Acoustics, Volume 112, Number 1,1990, pp. 84-92,

[0015] T. Pritz, “Transfer Function Method for Investigating the ComplexModulus of Acoustic Materials: Rod-Like Specimen,” Journal of Sound andVibration, Volume 81, 1982, pp. 359-376.

[0016] W. M. Madigosky and G. F. Lee, “Instrument for Measuring DynamicViscoelastic Properties,” U.S. Pat. No. 4,352,292, Oct. 5, 1982.

[0017] W. M. Madigosky and G. F. Lee, “Method for Measuring MaterialCharacteristics,” U.S. Pat. No. 4,418,573, Dec. 6, 1983.

[0018] W. Madigosky, “In Situ Dynamic Material Property MeasurementSystem,” U.S. Pat. No. 5,365,457, Nov. 15, 1994.

[0019] J. G. McDaniel, P. Dupont, and L. Salvino, “A Wave Approach toEstimating Frequency-Dependent Damping Under Transient Loading” Journalof Sound and Vibration, Volume 231(2), 2000, pp. 433-449.

[0020] J. Linjama and T. Lahti, “Measurement of Bending wave reflectionand Impedance in a Beam by the Structural Intensity Technique” Journalof Sound and Vibration, Volume 161(2), 1993, pp. 317-331.

[0021] L. Koss and D. Karczub, “Euler Beam Bending Wave SolutionPredictions of dynamic Strain Using Frequency Response Functions ”Journal of Sound and Vibration, Volume 184(2), 1995, pp. 229-244.

[0022] Measuring the flexural properties of beams is important becausethese parameters significantly contribute to the static and dynamicresponse of structures. In the past, resonant techniques have been usedto identify and measure longitudinal properties. These methods are basedon comparing the measured eigenvalues of a structure to predictedeigenvalues from a model of the same structure. The model of thestructure must have well-defined (typically closed form) eigenvalues forthis method to work. Additionally, resonant techniques only allowmeasurements at natural frequencies.

[0023] Comparison of analytical models to measured frequency responsefunctions is another method used to estimate stiffness and lossparameters of a structure. When the analytical model agrees with one ormore frequency response functions, the parameters used to calculate theanalytical model are considered accurate. If the analytical model isformulated using a numerical method, a comparison of the model to thedata can be difficult due to the dispersion properties of the materials.

[0024] Another method to measure stiffness and loss is to deform thematerial and measure the resistance to the indentation. This method canphysically damage the specimen if the deformation causes the sample toenter the plastic region of deformation.

SUMMARY OF THE INVENTION

[0025] Accordingly, one objective of the present invention is to measureflexural wavenumbers.

[0026] Another objective of the present invention is to measure flexuralwave propagation coefficients.

[0027] A further objective of the present invention is to measureYoung's modulus when the beam is undergoing transverse motion.

[0028] Yet another objective of the present invention is to measure theboundary stiffness and dampening values when the beam is vibratedtransversely.

[0029] The foregoing objects are attained by the method and system ofthe present invention. The present invention features a method ofdetermining structural properties of a flexural beam mounted to a base.The method comprises securing a plurality of accelerometers spacedapproximately equidistant from each other along a length of a beam. Oneaccelerometer can be secured to the base. An input is provided to thebeam. Seven frequency domain transfer functions of displacement aremeasured from the accelerometers secured to the beam. The flexuralwavenumber is estimated from the seven frequency domain transferfunctions.

[0030] The seven frequency domain transfer functions of displacementinclude the following equations:${T_{- 3} = {\frac{U_{- 3}\left( {{{- 3}\alpha},\omega} \right)}{V_{0}(\omega)} = {{A\quad {\cos \left( {3{\alpha\delta}} \right)}} - {B\quad {\sin \left( {3{\alpha\delta}} \right)}} + {C\quad {\cosh \left( {3{\alpha\delta}} \right)}} - {D\quad {\sinh \left( {3{\alpha\delta}} \right)}}}}},{T_{- 2} = {\frac{U_{- 2}\left( {{{- 2}\alpha},\omega} \right)}{V_{0}(\omega)} = {{A\quad {\cos \left( {2{\alpha\delta}} \right)}} - {B\quad {\sin \left( {2{\alpha\delta}} \right)}} + {C\quad {\cosh \left( {2{\alpha\delta}} \right)}} - {D\quad {\sinh \left( {2{\alpha\delta}} \right)}}}}},{T_{- 1} = {\frac{U_{- 1}\left( {{- \alpha},\omega} \right)}{V_{0}(\omega)} = {{A\quad {\cos ({\alpha\delta})}} - {B\quad {\sin ({\alpha\delta})}} + {C\quad {\cosh ({\alpha\delta})}} - {D\quad {\sinh ({\alpha\delta})}}}}},{T_{0} = {\frac{U_{0}\left( {0,\omega} \right)}{V_{0}(\omega)} = {A + C}}},{T_{1} = {\frac{U_{1}\left( {\delta,\omega} \right)}{V_{0}(\omega)} = {{A\quad {\cos ({\alpha\delta})}} + {B\quad {\sin ({\alpha\delta})}} + {C\quad {\cosh ({\alpha\delta})}} + {D\quad {\sinh ({\alpha\delta})}}}}},{T_{2} = {\frac{U_{2}\left( {{2\delta},\omega} \right)}{V_{0}(\omega)} = {{A\quad {\cos \left( {2{\alpha\delta}} \right)}} - {B\quad {\sin \left( {2{\alpha\delta}} \right)}} + {C\quad {\cosh \left( {2{\alpha\delta}} \right)}} + {D\quad {\sinh \left( {2{\alpha\delta}} \right)}}}}},{and}$${T_{3} = {\frac{U_{3}\left( {{3\delta},\omega} \right)}{V_{0}(\omega)} = {{A\quad {\cos \left( {3{\alpha\delta}} \right)}} + {B\quad {\sin \left( {3{\alpha\delta}} \right)}} + {C\quad {\cosh \left( {3{\alpha\delta}} \right)}} + {D\quad {\sinh \left( {3{\alpha\delta}} \right)}}}}},{and}$

[0031] The flexural wavenumber is determined using the followingequations: ${{Re}(\alpha)} = \left\{ {{\begin{matrix}{{{\frac{1}{2\delta}{Arc}\quad {\cos (s)}} + \frac{n\quad \pi}{2\delta}}} & {n\quad {even}} \\{{{\frac{1}{2\delta}{Arc}\quad {\cos \left( {- s} \right)}} + \frac{n\quad \pi}{2\delta}}} & {n\quad {odd}}\end{matrix}\quad {where}s} = {\left\lbrack {{Re}(\varphi)} \right\rbrack^{2} + \left\lbrack {{Im}(\varphi)} \right\rbrack^{2} - \sqrt{\left\{ {\left\lbrack {{Re}(\varphi)} \right\rbrack^{2} + \left\lbrack {{Im}(\varphi)} \right\rbrack^{2}} \right\}^{2} - \left\{ {{2\left\lbrack {{Re}(\varphi)} \right\rbrack}^{2} + {2\left\lbrack {{Im}(\varphi)} \right\rbrack}^{2} - 1} \right\}}}} \right.$

[0032] and said imaginary part comprises:${{Im}(\alpha)} = {\frac{1}{\delta}\log_{e}{\left\{ {\frac{{Re}(\varphi)}{\cos \left\lbrack {{{Re}(\alpha)}\delta} \right\rbrack} - \frac{{Im}(\varphi)}{\sin \left\lbrack {{{Re}(\alpha)}\delta} \right\rbrack}} \right\}.}}$

[0033] Using the flexural wavenumber and various equations disclosedwithin the present invention, the complex valued modulus of elasticitycan be determined at each frequency, as well as the wave propertycoefficient, and the boundary parameters.

[0034] Thus, this invention has the advantages that all measurements canbe calculated at every frequency that a transfer function measurement ismade. They do not depend on system resonance's or curve fitting totransfer functions. The calculation from transfer function measurementto calculation of all system parameters is exact, i.e., no errors areintroduced during this process. Furthermore, the measurements can becalculated without adverse consequences to the tested beam.

BRIEF DESCRIPTION OF THE DRAWINGS

[0035] These and other features and advantages of the present inventionwill be better understood in view of the following description of theinvention taken together with the drawings wherein:

[0036]FIG. 1 is a schematic block diagram of a conventional testingsystem including two springs and two dashpots attached to a shakertable;

[0037]FIG. 2 is a schematic block diagram of a conventional testingsystem including one spring and one dashpot attached to a shaker table;

[0038]FIG. 3 is a schematic block diagram of a conventional testingsystem including two springs and two dashpots, one of which is attachedto a shaker table;

[0039]FIG. 4 is a schematic block diagram of a conventional testingsystem wherein the beam is attached directly to a shaker table;

[0040]FIG. 5A is a graph of the magnitude of a typical transfer functionof a beam;

[0041]FIG. 5B is a graph of the phase angle of a typical transferfunction of a beam;

[0042]FIG. 6 is a graph of the function s versus frequency;

[0043]FIG. 7A is a graph of the real part of a flexural wavenumberversus frequency;

[0044]FIG. 7B is a graph of the imaginary part of a flexural wavenumberversus frequency;

[0045] FIGS. 8-11 are graphs of the wave propagation coefficients versusfrequency;

[0046]FIG. 12 is a graph of the real and imaginary parts of the Young'sModulus versus frequency;

[0047]FIG. 13 is a graph of the boundary conditions of the system shownin FIG. 1 versus frequency; and

[0048]FIG. 14 is a graph of the boundary conditions of the system shownin FIG. 2 versus frequency.

DESCRIPTION OF THE PREFERRED EMBODIMENT

[0049] The method and system, according to the present invention, isused to develop and measure complex flexural wavenumbers and thecorresponding wave propagation coefficients of a beam undergoingtransverse motion. An inverse method has been developed using seventransfer function measurements. These seven transfer functionmeasurements are combined to yield closed form values of flexuralwavenumber and wave propagation coefficients at any given testfrequency. Finally, Young's modulus, spring stiffnesses, dashpot dampingvalues, and boundary condition parameters, among other parameters, arecalculated from the flexural wavenumber and wave propagationcoefficients.

[0050] According to an exemplary test configuration 10, FIG. 1, a shakertable 12 initiates transverse motion 14 into a beam 16. The beam 16 isconnected to the shaker table 12 with a spring 18 and dashpot 20 at eachend 22. FIG. 1 represents a double translational spring and damper inputconfiguration. Other test configurations are also possible, includingthe shaker table 12 inputting energy into only one end 22 of the beam 16with the other end terminated to ground 24 directly, as shown in FIG. 2,or terminated to ground 24 with a spring 18 and dashpot 20, as shown inFIG. 3, or terminated to ground 24 and the shaker 12 directly, as shownin FIG. 4. FIG. 2 represents a single translational spring and damperinput configuration with the other end pinned. FIG. 3 represents asingle translational spring and damper input configuration with theother end having a translational spring and damper. FIG. 4 represents asingle pin input configuration with the other end pinned. Theseapproaches are intended for use when a beam 16 is to undergo motion inthe transverse direction 14. This system typically arises in cars,ships, aircraft, bridges, buildings and other common structures.

[0051] In any of the embodiments shown in FIGS. 1-4 sensors 26 such asaccelerators are positioned equally along beam 22. As discussed above, aminimum of seven such sensors 22 are required. Optionally, a referencesensor 28 can be joined to shaker table 12 to read the input motion 14.The input motion 14 can also be read directly from the shaker table 12controls.

[0052] For simplicity, the present invention will be described as itrelates to the derivation of the linear equations of motion of thesystem with a spring 18 and dashpot 20 boundary condition at each end22, but this is for exemplary purposes only, and is not intended to be alimitation.

[0053] The system model of the beam is the Bernoulli-Euler beam equationwritten as $\begin{matrix}{{{EI} = {{\frac{\partial^{4}{u\left( {x,t} \right)}}{\partial x^{4}} + {\rho \quad A_{b}\frac{\partial^{2}{u\left( {x,t} \right)}}{\partial t^{2}}}} = 0}},} & (1)\end{matrix}$

[0054] where x is the distance along the length of the beam in meters, tis time in seconds, u is the displacement of the beam in the(transverse) y direction in meters, E is the unknownfrequency-dependent, complex Young's modulus (N/m²), I is the moment ofinertia (m⁴), ρ is the density (kg/m³), and A_(b) is the cross-sectionalarea of the beam (m²). Implicit in equation (1) is the assumption thatplane sections remain planar during bending (or transverse motion).Additionally, Young's modulus, the moment of inertia, the density, andthe cross sectional area are constant across the entire length of thebeam. The displacement is modeled as a steady state response and isexpressed as

u(x,t)=U(x,ω)exp(iωt),  (2)

[0055] where ω is the frequency of excitation (rad/s), U(x,ω) is thetemporal Fourier transform of the transverse displacement, and i is thesquare root of −1. The temporal solution to equation (1), derived usingequation (2) and written in terms of trigonometric functions, is

U(x,ω)=A(ω)cos[α(ω)x]+B(ω)sin[α(ω)x]+C(ω)cosh[α(ω)x]+D(ω)sinh[α(ω)x]′  (3)

[0056] where A(ω), B(ω), C(ω), and D(ω) are wave propagationcoefficients and α(ω) is the flexural wavenumber given by$\begin{matrix}{{\alpha (\omega)} = {\left\lbrack \frac{\omega^{2}}{\left( {{{EI}/\rho}\quad A_{b}} \right)} \right\rbrack^{1/4}.}} & (4)\end{matrix}$

[0057] For brevity, the ω dependence is omitted from the wavepropagation coefficients and the flexural wavenumber during theremainder of the disclosure and α(ω) is references as α. Note thatequations (3) and (4) are independent of boundary conditions, and theinverse model developed in the next section does not need boundarycondition specifications. Boundary conditions are chosen, however, toshow that the boundary parameters can be estimated and to run arealistic simulation.

[0058] One of the most typical test configurations is the beam mountedto shock mounts on each end that are attached to a shaker table thatgenerates a vibrational input, as shown in FIG. 1. Using the middle ofthe beam as the coordinate system origin, these boundary conditions aremodeled as $\begin{matrix}{{\frac{\partial^{2}{u\left( {{{- L}/2},t} \right)}}{\partial x^{2}} = 0},} & (5) \\{{{{- {EI}}\frac{\partial^{2}{u\left( {{{- L}/2},t} \right)}}{\partial x^{3}}} = {{k_{1}\left\lbrack {{u\left( {{{- L}/2},t} \right)} - {v(t)}} \right\rbrack} + {c_{1}\left\lbrack {\frac{\partial{u\left( {{{- L}/2},t} \right)}}{\partial t} - \frac{\partial{v(t)}}{\partial t}} \right\rbrack}}},} & (6) \\{{\frac{\partial^{2}{u\left( {{L/2},t} \right)}}{\partial x^{2}} = 0},} & (7) \\{{{{- {EI}}\frac{\partial^{3}{u\left( {{L/2},t} \right)}}{\partial x^{3}}} = {{k_{2}\left\lbrack {{u\left( {{L/2},t} \right)} - {v(t)}} \right\rbrack} + {c_{2}\left\lbrack {\frac{\partial{u\left( {{L/2},t} \right)}}{\partial t} - \frac{\partial{v(t)}}{\partial t}} \right\rbrack}}},} & (8)\end{matrix}$

[0059] where

ν(t)=V ₀(ω)exp(iωt),tm (9)

[0060] which is the input into the system from the shaker table.

[0061] Inserting equation (3) into equation (5), (6), (7), (8), and (9)yields the solution to the wave propagation coefficients. Insertingthese back into equation (3) is the displacement of the system, and issometimes called the forward solution. The wave coefficient A is$\begin{matrix}{{A = \frac{A_{T}}{A_{B}}},} & (10)\end{matrix}$

[0062] where $\begin{matrix}{A_{T} = {{\left\lbrack {\left( {k_{1} + {\quad \omega \quad c_{1}}} \right) - \left( {k_{2} + {\quad \omega \quad c_{2}}} \right)} \right\rbrack \left( {{EI}\quad \alpha^{3}} \right){\cos \left( {\alpha \frac{L}{2}} \right)}{\cosh \left( {\alpha \frac{L}{2}} \right)}{\sinh \left( {\alpha \frac{L}{2}} \right)}} - {\quad{{{\left\lbrack {\left( {k_{1} + {\quad \omega \quad c_{1}}} \right) - \left( {k_{2} + {\quad \omega \quad c_{2}}} \right)} \right\rbrack \left( {{EI}\quad \alpha^{3}} \right){\sin \left( {\alpha \frac{L}{2}} \right)}{\cosh^{2}\left( {\alpha \frac{L}{2}} \right)}} - {4\left( {k_{1} + {\quad \omega \quad c_{1}}} \right)\left( {k_{2} + {\quad \omega \quad c_{2}}} \right){\sin \left( {\alpha \frac{L}{2}} \right)}{\cosh \left( {\alpha \frac{L}{2}} \right)}{\sinh \left( {\alpha \frac{L}{2}} \right)}}},}}}} & (11) \\{A_{B} = {{2\left( {{EI}\quad \alpha^{3}} \right)^{2}{\sin^{2}\left( {\alpha \frac{L}{2}} \right)}{\cosh \left( {\alpha \frac{L}{2}} \right)}} - {{2\left\lbrack {\left( {k_{1} + {\quad \omega \quad c_{1}}} \right) - \left( {k_{2} + {\quad \omega \quad c_{2}}} \right)} \right\rbrack}\left( {{EI}\quad \alpha^{3}} \right){\sin^{2}\left( {\alpha \frac{L}{2}} \right)}{\cosh \left( {\alpha \frac{L}{2}} \right)}{\sinh \left( {\alpha \frac{L}{2}} \right)}} - {2\left( {{EI}\quad \alpha^{3}} \right)^{2}{\cos^{2}\left( {\alpha \frac{L}{2}} \right)}{\sinh^{2}\left( {\alpha \frac{L}{2}} \right)}} - {{2\left\lbrack {\left( {k_{1} + {\quad \omega \quad c_{1}}} \right) - \left( {k_{2} + {\quad \omega \quad c_{2}}} \right)} \right\rbrack}\left( {{EI}\quad \alpha^{3}} \right){\cos \left( {\alpha \frac{L}{2}} \right)}{\sin \left( {\alpha \frac{L}{2}} \right)}{\sinh^{2}\left( {\alpha \frac{L}{2}} \right)}} + {{2\left\lbrack {\left( {k_{1} + {\quad \omega \quad c_{1}}} \right) - \left( {k_{2}\quad \omega \quad c_{2}} \right)} \right\rbrack}\left( {{EI}\quad \alpha^{3}} \right){\cos^{2}\left( {\alpha \frac{L}{2}} \right)}{\cosh \left( {\alpha \frac{L}{2}} \right)}{\sinh \left( {\alpha \frac{L}{2}} \right)}} - {{2\left\lbrack {\left( {k_{1} + {\quad \omega \quad c_{1}}} \right) - \left( {k_{2} + {\quad \omega \quad c_{2}}} \right)} \right\rbrack}\left( {{EI}\quad \alpha^{3}} \right){\cos \left( {\alpha \frac{L}{2}} \right)}{\sin \left( {\alpha \frac{L}{2}} \right)}{\cosh^{2}\left( {\alpha \frac{L}{2}} \right)}} - {8\left( {k_{1} + {\quad \omega \quad c_{1}}} \right)\left( {k_{2} + {\quad \omega \quad c_{2}}} \right){\cos \left( {\alpha \frac{L}{2}} \right)}{\sin \left( {\alpha \frac{L}{2}} \right)}{\cosh \left( {\alpha \frac{L}{2}} \right)}{\sinh \left( {\alpha \frac{L}{2}} \right)}}}} & (12)\end{matrix}$

[0063] and

[0064] The wave coefficient B is $\begin{matrix}{{B = \frac{B_{T}}{B_{B}}},} & (13)\end{matrix}$

[0065] where $\begin{matrix}{B_{T} = {{{- \left\lbrack {\left( {k_{1} + {\quad \omega \quad c_{1}}} \right) + \left( {k_{2}\quad \omega \quad c_{2}} \right)} \right\rbrack}\left( {{EI}\quad \alpha^{3}} \right){\sin \left( {\alpha \frac{L}{2}} \right)}{\cosh \left( {\alpha \frac{L}{2}} \right)}{\sinh \left( {\alpha \frac{L}{2}} \right)}} - {\quad{{\left\lbrack {\left( {k_{1} + {\quad \omega \quad c_{1}}} \right) + \left( {k_{2} + {\quad \omega \quad c_{2}}} \right)} \right\rbrack \left( {{EI}\quad \alpha^{3}} \right){\cos \left( {\alpha \frac{L}{2}} \right)}{\sinh^{2}\left( {\alpha \frac{L}{2}} \right)}},}}}} & (14) \\{B_{B} = {{2\left( {{EI}\quad \alpha^{3}} \right)^{2}{\cos^{2}\left( {\alpha \frac{L}{2}} \right)}{\sinh^{2}\left( {\alpha \frac{L}{2}} \right)}} - {{2\left\lbrack {\left( {k_{1} + {\quad \omega \quad c_{1}}} \right) - \left( {k_{2} + {\quad \omega \quad c_{2}}} \right)} \right\rbrack}\left( {{EI}\quad \alpha^{3}} \right){\cos^{2}\left( {\alpha \frac{L}{2}} \right)}{\cosh \left( {\alpha \frac{L}{2}} \right)}{\sinh \left( {\alpha \frac{L}{2}} \right)}} - {2\left( {{EI}\quad \alpha^{3}} \right)^{2}{\sin^{2}\left( {\alpha \frac{L}{2}} \right)}{\cosh^{2}\left( {\alpha \frac{L}{2}} \right)}} + {{2\left\lbrack {\left( {k_{1} + {\quad \omega \quad c_{1}}} \right) - \left( {k_{2} + {\quad \omega \quad c_{2}}} \right)} \right\rbrack}\left( {{EI}\quad \alpha^{3}} \right){\cos \left( {\alpha \frac{L}{2}} \right)}{\sin \left( {\alpha \frac{L}{2}} \right)}{\cosh^{2}\left( {\alpha \frac{L}{2}} \right)}} + {{2\left\lbrack {\left( {k_{1} + {\quad \omega \quad c_{1}}} \right) - \left( {k_{2} + {\quad \omega \quad c_{2}}} \right)} \right\rbrack}\left( {{EI}\quad \alpha^{3}} \right){\sin^{2}\left( {\alpha \frac{L}{2}} \right)}{\cosh \left( {\alpha \frac{L}{2}} \right)}{\sinh \left( {\alpha \frac{L}{2}} \right)}} + {{2\left\lbrack {\left( {k_{1} + {\quad \omega \quad c_{1}}} \right) - \left( {k_{2} + {\quad \omega \quad c_{2}}} \right)} \right\rbrack}\left( {{EI}\quad \alpha^{3}} \right){\cos \left( {\alpha \frac{L}{2}} \right)}{\sin \left( {\alpha \frac{L}{2}} \right)}{\sinh^{2}\left( {\alpha \frac{L}{2}} \right)}} + {8\left( {k_{1} + {\quad \omega \quad c_{1}}} \right)\left( {k_{2} + {\quad \omega \quad c_{2}}} \right){\cos \left( {\alpha \frac{L}{2}} \right)}{\sin \left( {\alpha \frac{L}{2}} \right)}{\cosh \left( {\alpha \frac{L}{2}} \right)}{\sinh \left( {\alpha \frac{L}{2}} \right)}}}} & (15)\end{matrix}$

[0066] and

[0067] The wave coefficient C is $\begin{matrix}{{C = \frac{C_{T}}{A_{B}}},} & (16)\end{matrix}$

[0068] where $\begin{matrix}{C_{T} = {{\left\lbrack {\left( {k_{1} + {i\quad \omega \quad c_{1}}} \right) - \left( {k_{2} + {i\quad \omega \quad c_{2}}} \right)} \right\rbrack \left( {{EI}\quad \alpha^{3}} \right){\cos^{2}\left( {\alpha \frac{L}{2}} \right)}{\sinh \left( {\alpha \frac{L}{2}} \right)}} - {\left. \quad{{\left\lbrack {\left( {k_{1} + {i\quad \omega \quad c_{1}}} \right) - \left( {k_{2} + {i\quad \omega \quad c_{2}}} \right)} \right\rbrack \left( {{EI}\quad \alpha^{3}} \right){\cos \left( {\alpha \frac{L}{2}} \right)}{\sin \left( {\alpha \frac{L}{2}} \right)}{\cosh \left( {\alpha \frac{L}{2}} \right)}} - {4\left( {k_{1} + {i\quad \omega \quad c_{1}}} \right)\left( {k_{2} + {i\quad \omega \quad c_{2}}} \right)}} \right\rbrack {\cos \left( {\alpha \frac{L}{2}} \right)}{\sin \left( {\alpha \frac{L}{2}} \right)}{{\sinh \left( {\alpha \frac{L}{2}} \right)}.}}}} & (17)\end{matrix}$

[0069] The wave coefficient D is $\begin{matrix}{{D = \frac{D_{T}}{B_{B}}},} & (18)\end{matrix}$

[0070] where $\begin{matrix}{D_{T} = {{{- \left\lbrack {\left( {k_{1} + {i\quad \omega \quad c_{1}}} \right) + \left( {k_{2} + {i\quad \omega \quad c_{2}}} \right)} \right\rbrack}\left( {{EI}\quad \alpha^{3}} \right){\sin^{2}\left( {\alpha \frac{L}{2}} \right)}{\cosh \left( {\alpha \frac{L}{2}} \right)}} - {\quad{\left\lbrack {\left( {k_{1} + {i\quad \omega \quad c_{1}}} \right) + \left( {k_{2} + {i\quad \omega \quad c_{2}}} \right)} \right\rbrack \left( {{EI}\quad \alpha^{3}} \right){\cos \left( {\alpha \frac{L}{2}} \right)}{\sin \left( {\alpha \frac{L}{2}} \right)}{{\sinh \left( {\alpha \frac{L}{2}} \right)}.}}}}} & (19)\end{matrix}$

[0071] These coefficients are used for the simulation below. If the beammodel corresponds to FIGS. 2, 3, or 4, then the boundary conditionsgiven in equations (5)-(8) change slightly as do the wave propagationcoefficients.

[0072] Equation (3) has five unknowns and is nonlinear with respect tothe unknown flexural wavenumber. It will be shown that using sevenindependent, equally spaced measurements, that the five unknowns can beestimated with closed form solutions. Furthermore, in the next section,it will be shown that the components that comprise the beams mountingsystem can also be estimated. Seven frequency domain transfer functionsof displacement are now measured. These consist of the measurement atsome location divided by a common measurement. Typically this would bean accelerometer at a measurement location and an accelerometer at thebase of a shaker table. These seven measurements are set equal thetheoretical expression given in equation (3) and are listed as$\begin{matrix}{{T_{- 3} = {\frac{U_{- 3}\left( {{{- 3}\quad \delta},\omega} \right)}{V_{0}(\omega)} = {{A\quad {\cos \left( {3\quad \alpha \quad \delta} \right)}} - {B\quad {\sin \left( {3\quad \alpha \quad \delta} \right)}} + {C\quad {\cosh \left( {3\quad \alpha \quad \delta} \right)}} - {D\quad {\sinh \left( {3\quad {\alpha\delta}} \right)}}}}},} & (20) \\{{T_{- 2} = {\frac{U_{- 2}\left( {{{- 2}\quad \delta},\omega} \right)}{V_{0}(\omega)} = {{A\quad {\cos \left( {2\quad \alpha \quad \delta} \right)}} - {B\quad {\sin \left( {2\quad \alpha \quad \delta} \right)}} + {C\quad {\cosh \left( {2\quad \alpha \quad \delta} \right)}} - {D\quad {\sinh \left( {2\quad {\alpha\delta}} \right)}}}}},} & (21) \\{{T_{- 1} = {\frac{U_{- 1}\left( {{- \quad \delta},\omega} \right)}{V_{0}(\omega)} = {{A\quad {\cos \left( {\alpha \quad \delta} \right)}} - {B\quad {\sin \left( {\alpha \quad \delta} \right)}} + {C\quad {\cosh \left( {\alpha \quad \delta} \right)}} - {D\quad {\sinh ({\alpha\delta})}}}}},} & (22) \\{{T_{0} = {\frac{U_{0}\left( {0,\omega} \right)}{V_{0}(\omega)} = {A + C}}},} & (23) \\{{T_{1} = {{\frac{U_{1}\left( {\delta,\omega} \right)}{V_{0}(\omega)}A\quad {\cos ({\alpha\delta})}} + {B\quad {\sin \left( {\alpha \quad \delta} \right)}} + {C\quad {\cosh ({\alpha\delta})}} + {D\quad {\sinh \left( {\alpha \quad \delta} \right)}}}},} & (24) \\{{T_{2} = {{\frac{U_{2}\left( {{2\delta},\omega} \right)}{V_{0}(\omega)}A\quad {\cos \left( {2{\alpha\delta}} \right)}} + {B\quad {\sin \left( {2\alpha \quad \delta} \right)}} + {C\quad {\cosh \left( {2{\alpha\delta}} \right)}} + {D\quad {\sinh \left( {2\alpha \quad \delta} \right)}}}},{and}} & (25) \\{{T_{3} = {{\frac{U_{3}\left( {{3\delta},\omega} \right)}{V_{0}(\omega)}A\quad {\cos \left( {3{\alpha\delta}} \right)}} + {B\quad {\sin \left( {3\alpha \quad \delta} \right)}} + {C\quad {\cosh \left( {3{\alpha\delta}} \right)}} + {D\quad {\sinh \left( {3\alpha \quad \delta} \right)}}}},} & (26)\end{matrix}$

[0073] where δ is the sensor to sensor separation distance (m) and V₀(ω)is the reference measurement. Note that the units of the transferfunctions given in equations (20)-(26) are dimensionless.

[0074] Equation (22) is now subtracted from equation (24), equation (21)is subtracted from equation (25), and equation (20) is subtracted fromequation (26), yielding the following three equations: $\begin{matrix}{{{{B\quad \sin \left( {\alpha \quad \delta} \right)} + {D\quad {\sinh \left( {\alpha \quad \delta} \right)}}} = \frac{T_{1} - T_{- 1}}{2}},} & (27) \\{{{{B\quad {\sin \left( {2\alpha \quad \delta} \right)}} + {D\quad {\sinh \left( {2\alpha \quad \delta} \right)}}} = \frac{T_{2} - T_{- 2}}{2}},{and}} & (28) \\{{{B\quad {\sin \left( {3\alpha \quad \delta} \right)}} + {D\quad {\sinh \left( {3\alpha \quad \delta} \right)}}} = {\frac{T_{3} - T_{- 3}}{2}.}} & (29)\end{matrix}$

[0075] Equations (27), (28), and (29) are now combined to give$\begin{matrix}{{{{\cosh ({\alpha\delta})}{\cos ({\alpha\delta})}} - {\left\lbrack \quad \frac{T_{2} - T_{- 2}}{2\left( {T_{1} - T_{- 1}} \right)} \right\rbrack\left\lbrack \quad {{\cosh ({\alpha\delta})} + {\cos ({\alpha\delta})}} \right\rbrack} + \left\lbrack \quad \frac{T_{3} - T_{- 3} + T_{1} - T_{- 1}}{4\left( {T_{1} - T_{- 1}} \right)} \right\rbrack} = 0.} & (30)\end{matrix}$

[0076] Equation (22) is now added to equation (24) and equation (21) isadded to equation (25), yielding the following two equations:$\begin{matrix}{{{{A\quad {\cos ({\alpha\delta})}} + {C\quad {\cosh ({\alpha\delta})}}} = \frac{T_{1} + T_{- 1}}{2}},} & (31)\end{matrix}$

[0077] and $\begin{matrix}{{{A\quad {\cos \left( {2{\alpha\delta}} \right)}} + {C\quad {\cosh \left( {2{\alpha\delta}} \right)}}} = {\frac{T_{2} + T_{- 2}}{2}.}} & (32)\end{matrix}$

[0078] Equations (23), (31), and (32) are now combined to yield thefollowing equation: $\begin{matrix}{{{{\cosh ({\alpha\delta})}{\cos ({\alpha\delta})}} - {\left\lbrack \quad \frac{T_{2} - T_{- 2}}{2T_{0}} \right\rbrack\left\lbrack \quad {{\cosh ({\alpha\delta})} + {\cos ({\alpha\delta})}} \right\rbrack} + \left\lbrack \quad \frac{T_{2} + T_{- 2} + {2T_{0}}}{\left. {4T_{0}} \right)} \right\rbrack} = 0.} & (33)\end{matrix}$

[0079] Equation (30) and (33) are now combined, and the result is abinomial expression with respect to the cosine function, and is writtenas

a cos²(αδ)+b cos(αδ)+c=0,  (34)

[0080] where

a=4T ₁ ²−4T ⁻¹ ²+4T ⁻² T ₀−4T ₀ T ₂,  (35)

b=2T ⁻² T ⁻¹−2T ⁻² T ₁+2T ⁻¹ T ₀2T ₀ T ₁+2T ⁻¹ T ₂−2T ₁ T ₂+2T ₀ T ₃−2T⁻³ T ₀,  (36)

[0081] and

c=T ⁻¹ ² −T ₁ ² +T ₂ ² −T ⁻² ² +T ⁻³ T ⁻¹ −T ⁻¹ T ₃ +T ⁻³ T ₁ −T ₁ T₃+2T ₀ T ₂−2T ⁻² T ₀.  (37)

[0082] Equation (34) is now solved using $\begin{matrix}{{{\cos ({\alpha\delta})} = {\frac{{- b} \pm \sqrt{b^{2} - {4\quad {ac}}}}{2a} = \varphi}},} & (38)\end{matrix}$

[0083] where φ is typically a complex number. Equation (38) is twosolutions to equation (34). One, however, will have an absolute valueless than one and that is the root that is further manipulated. Theinversion of equation (38) allows the complex flexural wavenumber α tobe solved as a function of φ at every frequency in which a measurementis made. The solution to the real part of α is $\begin{matrix}{{{Re}(\alpha)} = \left\{ {\begin{matrix}{{\frac{1}{2\delta}{Arc}\quad {\cos (s)}} + \frac{n\quad \pi}{2\delta}} & {n\quad {even}} \\{{\frac{1}{2\delta}{Arc}\quad {\cos \left( {- s} \right)}} + \frac{n\quad \pi}{2\delta}} & {n\quad {odd}}\end{matrix},} \right.} & (39)\end{matrix}$

[0084] where $\begin{matrix}{{s = {\left\lbrack {{Re}(\varphi)} \right\rbrack^{2} + \left\lbrack {{Im}(\varphi)} \right\rbrack^{2} - \sqrt{\left\{ {\left\lbrack {{Re}(\varphi)} \right\rbrack^{2} + \left\lbrack {{Im}(\varphi)} \right\rbrack^{2}} \right\}^{2} - \left\{ {{2\left\lbrack {{Re}(\varphi)} \right\rbrack}^{2} - {2\left\lbrack {{Im}(\varphi)} \right\rbrack}^{2} - 1} \right\}}}},} & (40)\end{matrix}$

[0085] n is a non-negative integer and the capital A denotes theprincipal value of the inverse cosine function. The value of n isdetermined from the function s, which is a periodically varying cosinefunction with respect to frequency. At zero frequency, n is 0. Everytime s cycles through π radians (180 degrees), n is increased by 1. Whenthe solution to the real part of α is found, the solution to theimaginary part of α is then written as $\begin{matrix}{{{Im}(\alpha)} = {\frac{1}{\delta}\log_{e}{\left\{ {\frac{{Re}(\varphi)}{\cos \left\lbrack {{{Re}(\alpha)}\delta} \right\rbrack} - \frac{{Im}(\varphi)}{\sin \left\lbrack {{{Re}(\alpha)}\delta} \right\rbrack}} \right\}.}}} & (41)\end{matrix}$

[0086] Once the real and imaginary parts of wavenumber α are known, thecomplex valued modulus of elasticity can be determined at each frequencywith $\begin{matrix}{{E(\omega)} = {{{{Re}\left\lbrack {E(\omega)} \right\rbrack} + {i\quad {{Im}\left\lbrack {E(\omega)} \right\rbrack}}} = {\frac{\rho \quad A_{b}\omega^{2}}{{I\left\lbrack {{{Re}(\alpha)} + {i\quad {{Im}(\alpha)}}} \right\rbrack}^{4}}.}}} & (42)\end{matrix}$

[0087] assuming that the density, area, and moment of inertia of thebeam are known. Equations (20)-(42) produce an estimate Young's modulusat every frequency in which a measurement is conducted.

[0088] Additionally, combining equations (27) and (28) yields$\begin{matrix}{B = \frac{{2\left( {T_{1} - T_{- 1}} \right){\cosh ({\alpha\delta})}} - \left( {T_{2} - T_{- 2}} \right)}{4\quad {{\sin ({\alpha\delta})}\left\lbrack {{\cosh ({\alpha\delta})} - {\cos ({\alpha\delta})}} \right\rbrack}}} & (43)\end{matrix}$

[0089] and $\begin{matrix}{D = {\frac{\left( {T_{2} - T_{- 2}} \right) - {2\left( {T_{1} - T_{- 1}} \right){\cos ({\alpha\delta})}}}{4\quad {{\sinh ({\alpha\delta})}\left\lbrack {{\cosh ({\alpha\delta})} - {\cos ({\alpha\delta})}} \right\rbrack}}.}} & (44)\end{matrix}$

[0090] Combining equations (23) and (31) yields $\begin{matrix}{A = \frac{{2T_{0}{\cosh ({\alpha\delta})}} - \left( {T_{1} + T_{- 1}} \right)}{2\left\lbrack {{\cosh ({\alpha\delta})} - {\cos ({\alpha\delta})}} \right\rbrack}} & (45)\end{matrix}$

[0091] and $\begin{matrix}{C = {\frac{\left( {T_{1} + T_{- 1}} \right) - {2T_{0}{\cos ({\alpha\delta})}}}{2\left\lbrack {{\cosh ({\alpha\delta})} - {\cos ({\alpha\delta})}} \right\rbrack}.}} & (46)\end{matrix}$

[0092] Equations (43)-(46) are the estimates of the complex wavepropagation coefficients. These are normally considered less importantthan the estimate of the flexural wavenumber. It will be shown, however,that these coefficients can be used to estimate the boundary conditionparameters of the beam.

[0093] Inserting equations (2), (3), (4), and (9) into equation (6) andsolving for the boundary parameters at x=−L/2 yields $\begin{matrix}{k_{1} = {{Re}\left\{ \frac{\left( {{EI}\quad \alpha^{3}} \right)\left\lfloor {{A\quad {\sin \left( {\alpha \frac{L}{2}} \right)}} + {B\quad {\cos \left( {\alpha \frac{L}{2}} \right)}} + {C\quad {\sinh \left( {\alpha \frac{L}{2}} \right)}} - {D\quad {\cosh \left( {\alpha \frac{L}{2}} \right)}}} \right\rfloor}{\left\lbrack {{A\quad {\cos \left( {\alpha \frac{L}{2}} \right)}} - {B\quad {\sin \left( {\alpha \frac{L}{2}} \right)}} + {C\quad {\cosh \left( {\alpha \frac{L}{2}} \right)}} - {D\quad {\sinh \left( {\alpha \frac{L}{2}} \right)}} - 1} \right\rbrack} \right\}}} & (47)\end{matrix}$

[0094] and $\begin{matrix}{c_{1} = {\frac{1}{\omega}{Im}{\left\{ \frac{\left( {{EI}\quad \alpha^{3}} \right)\left\lbrack {{A\quad {\sin \left( {\alpha \frac{L}{2}} \right)}} + {B\quad {\cos \left( {\alpha \frac{L}{2}} \right)}} + {C\quad {\sinh \left( {\alpha \frac{L}{2}} \right)}} - {D\quad {\cosh \left( {\alpha \frac{L}{2}} \right)}}} \right\rbrack}{\left\lbrack {{A\quad {\cos \left( {\alpha \frac{L}{2}} \right)}} - {B\quad {\sin \left( {\alpha \frac{L}{2}} \right)}} + {C\quad {\cosh \left( {\alpha \frac{L}{2}} \right)}} - {D\quad {\sinh \left( {\alpha \frac{L}{2}} \right)}} - 1} \right\rbrack} \right\}.}}} & (48)\end{matrix}$

[0095] Similarly, inserting equations (2), (3), (4), and (9) intoequation (8) and solving for the boundary parameters at x=L/2 yields$\begin{matrix}{k_{2} = {{Re}\left\{ \frac{\begin{matrix}{\left( {{- {EI}}\quad \alpha^{3}} \right)\left\lbrack {{A\quad {\sin \left( {\alpha \frac{L}{2}} \right)}} - {B\quad \cos \left( {\alpha \frac{L}{2}} \right)} +} \right.} \\\left. {{C\quad {\sinh \left( {\alpha \frac{L}{2}} \right)}} + {D\quad {\cosh \left( {\alpha \frac{L}{2}} \right)}}} \right\rbrack\end{matrix}}{\left\lbrack {{A\quad {\cos \left( {\alpha \frac{L}{2}} \right)}} + {B\quad {\sin \left( {\alpha \frac{L}{2}} \right)}} + {C\quad {\cosh \left( {\alpha \frac{L}{2}} \right)}} + {D\quad {\sinh \left( {\alpha \frac{L}{2}} \right)}} - 1} \right\rbrack} \right\}}} & (49)\end{matrix}$

[0096] and $\begin{matrix}{c_{2} = {\frac{1}{\omega}{Im}{\left\{ \frac{\begin{matrix}{\left( {{- {EI}}\quad \alpha^{3}} \right)\left\lbrack {{A\quad {\sin \left( {\alpha \frac{L}{2}} \right)}} - {B\quad \cos \left( {\alpha \frac{L}{2}} \right)} +} \right.} \\\left. {{C\quad \sinh \left( {\alpha \frac{L}{2}} \right)} + {D\quad {\cosh \left( {\alpha \frac{L}{2}} \right)}}} \right\rbrack\end{matrix}}{\left\lbrack {{A\quad {\cos \left( {\alpha \frac{L}{2}} \right)}} + {B\quad {\sin \left( {\alpha \frac{L}{2}} \right)}} + {C\quad {\cosh \left( {\alpha \frac{L}{2}} \right)}} + {D\quad {\sinh \left( {\alpha \frac{L}{2}} \right)}} - 1} \right\rbrack} \right\}.}}} & (50)\end{matrix}$

[0097] Thus, once the flexural wavenumber and wave coefficients areestimated, the properties of the springs and dashpots at the boundariescan be calculated.

[0098] Numerical simulations conducted to determine the effectiveness ofthis method use the following parameters to define a baseline problem:Re(E)=(3·10¹⁰+10⁷f) N/m², Im(E)=(3·10⁹+10⁶f) N/m², ρ=5000 kg/m³,A_(b)=0.02 m², I=6.67×10⁻⁵ m⁴, L=3 m, δ=0.5 m, k₁=50000 N/m, c₁=4000N·s/m, k₂=60000 N/m, and c₂=5000 N·s/m where f is frequency in Hz. FIGS.5A and 5B represent a typical transfer function of the beam displacementmeasured at x=0 m, which is the middle of the beam, divided by basedisplacement. The top plot, FIG. 5A, is the magnitude versus frequencyand the bottom plot, FIG. 5B, is the phase angle versus frequency. Thisfigure was constructed by inserting the above parameters into equations(3), (4), (10), (11), (12), (13), (14), (15), (16), (17), (18), and (19)and calculating the solution (a forward model).

[0099]FIG. 6 graphs the function s versus frequency. It was calculatedby inserting the left-hand side of equations (20)-(26) into equations(34)-(40) and represents the first step of the inverse methodcalculations. FIGS. 7A and 7B represent the flexural wavenumber versusfrequency. The top plot, FIG. 7A, is the real part and the bottom plot,FIG. 7B, is the imaginary part. The values created using equation (4)(the forward solution) are shown as solid lines and the valuescalculated (or estimated) using equations (34)-(41) (the inversesolution) are shown with x's and o's. Note that there is total agreementamong the forward and inverse solutions. FIGS. 8-11 are the wavepropagation coefficients A, B, C, and D versus frequency, respectively.The top plots are the magnitudes and the bottom plots are the phaseangles. The values created using equation (10)-(19) (the forwardsolution) are shown as solid lines and the values calculated usingequations (43)-(46) (the inverse solution) are shown with x's and o's.FIG. 12A and FIG. 12B graph the real and imaginary parts of Young'smodulus versus frequency. The actual values are shown as solid lines andthe values calculated using equation (42) are shown with x's and o's.FIG. 13 is the boundary condition parameters of mount one versusfrequency. The top plot is the stiffness and the bottom plot is thedamping. The actual values are shown as solid lines and the valuescalculated using equations (47) and (48) are shown with x's and o's.FIG. 14 is the boundary condition parameters of mount two versusfrequency. The top plot, FIG. 14A, is the stiffness and the bottom plot,FIG. 14B, is the damping. The actual values are shown as solid lines andthe values calculated using equations (49) and (50) are shown with x'sand o's.

[0100] In light of the above, it is therefore understood that within thescope of the appended claims, the invention may be practiced otherwisethan as specifically described.

What is claimed is:
 1. A method of determining structural properties ofa flexural beam comprising the steps of: securing a plurality ofaccelerometers spaced approximately equidistant from each other along alength of said beam; providing a vibrational input to said beam;measuring seven frequency domain transfer functions of displacement; andestimating a flexural wavenumber from said seven frequency domaintransfer functions.
 2. The method of claim 1 wherein said sevenfrequency domain transfer functions comprise: $\begin{matrix}{T_{- 3} = {\frac{U_{- 3}\left( {{{- 3}\quad \delta},\omega} \right)}{V_{0}(\omega)} = {{A\quad \cos \left( {3\alpha \quad \delta} \right)} -}}} \\{{{{B\quad \sin \left( {3{\alpha\delta}} \right)} + {C\quad \cosh \left( {3\alpha \quad \delta} \right)} - {D\quad \sinh \left( {3\alpha \quad \delta} \right)}},}} \\{T_{- 2} = {\frac{U_{- 2}\left( {{{- 2}\quad \delta},\omega} \right)}{V_{0}(\omega)} = {{A\quad \cos \left( {2\alpha \quad \delta} \right)} -}}} \\{{{{B\quad \sin \left( {2{\alpha\delta}} \right)} + {C\quad {\cosh \left( {2\alpha \quad \delta} \right)}} - {D\quad {\sinh \left( {2\alpha \quad \delta} \right)}}},}} \\{T_{- 1} = {\frac{U_{- 1}\left( {{- \delta},\omega} \right)}{V_{0}(\omega)} = {{A\quad \cos \left( {\alpha \quad \delta} \right)} -}}} \\{{{{B\quad \sin ({\alpha\delta})} + {C\quad {\cosh \left( {\alpha \quad \delta} \right)}} - {D\quad {\sinh \left( {\alpha \quad \delta} \right)}}},}} \\{{T_{0} = {\frac{U_{0}\left( {0,\omega} \right)}{V_{0}(\omega)} = {A + C}}},} \\{{T_{1} = {\frac{U_{1}\left( {\delta,\omega} \right)}{V_{0}(\omega)} = {{A\quad {\cos \left( {\alpha \quad \delta} \right)}} + {B\quad {\sin ({\alpha\delta})}} + {C\quad {\cosh \left( {\alpha \quad \delta} \right)}} + {D\quad {\sinh \left( {\alpha \quad \delta} \right)}}}}},} \\{T_{2} = {\frac{U_{2}\left( {{2\quad \delta},\omega} \right)}{V_{0}(\omega)} = {{A\quad \cos \left( {2\alpha \quad \delta} \right)} +}}} \\{{{{B\quad \sin \left( {2{\alpha\delta}} \right)} + {C\quad {\cosh \left( {2\alpha \quad \delta} \right)}} + {D\quad {\sinh \left( {2\alpha \quad \delta} \right)}}},{and}}} \\{T_{3} = {\frac{U_{3}\left( {{3\quad \delta},\omega} \right)}{V_{0}(\omega)} = {{A\quad \cos \left( {3\alpha \quad \delta} \right)} -}}} \\{{{B\quad \sin \left( {3{\alpha\delta}} \right)} + {C\quad {\cosh \left( {3\alpha \quad \delta} \right)}} + {D\quad {{\sinh \left( {3\alpha \quad \delta} \right)}.}}}}\end{matrix}$


3. The method of claim 2 further comprising the step of securing atleast one accelerometer to said base.
 4. The method of claim 2 furthercomprising securing said beam to a shaker table using a spring and adashpot disposed at both a first and a second end of said beam;
 5. Themethod of claim 4 further comprising the step of securing at least oneaccelerometer to said base.
 6. The method of claim 2 further comprising:securing a first end of said beam to a shaker table using a spring and adashpot; and securing a second end of said beam to a fixed object by apinned connection.
 7. The method of claim 6 further comprising the stepof securing at least one accelerometer to said base.
 8. The method ofclaim 2 further comprising the steps of: securing a first end of saidbeam to a shaker table using a spring and a dashpot; and securing asecond end of said beam to a fixed object using a spring and a dashpot.9. The method of claim 8 further comprising the step of securing atleast one accelerometer to said base.
 10. The method of claim 2 furthercomprising the steps of: securing a first end of said beam directly to ashaker table; and securing a second end of said beam to a fixed objectby a pinned connection.
 11. The method of claim 10 further comprisingthe step of securing at least one accelerometer to said base.
 12. Themethod of claim 1 further comprising the step of determining a complexvalued modulus of elasticity at each frequency using said flexuralwavenumber.
 13. The method of claim 1 further comprising the step ofdetermining wave property coefficient using said flexural wavenumber.